Related papers: On the tensor Permutation Matrices

We show how to construct relevant families of matrix product operators in one and higher dimensions. Those form the building blocks for the numerical simulation methods based on matrix product states and projected entangled pair states. In…

The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic…

We study permutation type solutions to n-simplex equations, that is, solutions whose R matrix can be written as a product of delta- functions depending linearly on the indices. With this ansatz the D^{n(n+1)} equations of the n-simplex…

We study the concept of frame in tensor product of n-Hilbert spaces as tensor product of n-Hilbert spaces is again a n-Hilbert space. We generalize some of the known results about bases to frames in this new Hilbert space. A relationship…

Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the $20^{\text{th}}$ century, they have since then spread to…

Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for…

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In…

We introduce tensor generalized bilateral inverses (TGBIs) under the Einstein tensor product as an extension of generalized bilateral inverses (GBIs) in the matrix environment. Moreover, the TBGI class includes so far considered composite…

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear…

We consider the problem of determining which matrices are permutable to be supmodular. We show that for small dimensions any matrix is permutable by a universal permutation or by a pair of permutations, while for higher dimensions no…

On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we…

Square matrices represent linear self-maps of vector spaces, and their eigenpoints are the fixed points of the induced map on projective space. Likewise, polynomial self-maps of a projective space are represented by tensors. We study the…

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…

Base on some simple facts of Hadamard product, characterizations of positive definite preserving linear transformations on real symmetric matrix spaces with an additional assumption "$\ra T(E_{ii})=1, i=1,2,..., n$" or "$T(A)>0\to A> 0$",…

We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…

We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible…

Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its…

We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We…

We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these…

We construct two sets of representations of the Dirac equation. They transform themselves from one to another in multiplying by the tensor commutation matrix (TCM) $2\otimes 2$. The Gauss matrix can lead us to the Cholesky decomposition.…